If I start a fire with the woodstove, what happens?
The owner’s insurer pays the owner and can then pursue you for the cost. That figure is the building rather than your belongings, which is why the liability limit on a renters policy matters far more than the contents limit. It is the strongest argument for not carrying the minimum.
Who is supposed to clean the chimney?
Usually the owner, as part of maintaining the property, with day-to-day use being yours. Leases are frequently silent on it, and that silence is what gets argued about after a fire. Ask in writing at the start of the tenancy.
Is there anything simple I should do?
Put ash in a metal container with a lid, away from the building, and leave it for days before disposing of it. Ash into a plastic bin or onto a deck causes fires hours after the stove is out, and it is the most preventable version of this claim.